Contents

Motivation

Consider some vectors in a vector space \( V \) over \( \mathbb{F} \). We would like to define some concepts such as length of a vector or angle between two vectors, both of which are not intrinsic properties of the space itself. We notice that scalar multiplying a vector \( \vec{v} \) by a factor of \( c \) multiplies its length by \( c \), and forming the sum \( \vec{v} + \vec{w} \) causes the norm to change from \( \sqrt{\sum_i v_i^2} \) to \( \sqrt{\sum_i v_i^2 + \sum_i w_i^2 + 2 \sum_i v_i w_i} \). Let us abstract the concept. Such a function \( L: V \times W \to \mathbb{F} \) taking in two inputs has the property of being linear with respect to each input; that is, \( L(c \vec{v} + \vec{a}, d \vec{w} + \vec{b} ) = cdL(\vec{v}, \vec{w} ) + cL(\vec{v}, \vec{b}) + dL(\vec{a}, \vec{w}) + L(\vec{b}, \vec{w}) \). For a function of \( n \) inputs linear in each input, this function is called \( n \) linear. For \( n =2, V = W \), such a function is called an inner product. Inner products will be used to develop the ideas of the magnitude of a vector and the angles between two vectors in Euclidean spaces as well as some other more abstract ideas.